Lov\'asz-Schrijver SDP-operator, near-perfect graphs and near-bipartite graphs

TL;DR

This paper investigates the Lovász-Schrijver $LS_+$ operator's role in characterizing certain graph classes, aiming to identify graphs where the stable set polytope is obtained in one step, and explores relationships among $LS_+$-perfect, near-bipartite, and full-support-perfect graphs.

Contribution

It advances the understanding of $LS_+$-perfect graphs by establishing new relationships with near-bipartite and full-support-perfect graphs, moving toward a full combinatorial characterization.

Findings

Established a new relationship between $LS_+$-perfect and near-bipartite graphs.

Introduced the concept of full-support-perfect graphs.

Progressed towards a combinatorial characterization of $LS_+$-perfect graphs.

Abstract

We study the Lov\'asz-Schrijver lift-and-project operator ($LS_+$) based on the cone of symmetric, positive semidefinite matrices, applied to the fractional stable set polytope of graphs. The problem of obtaining a combinatorial characterization of graphs for which the $LS_+$-operator generates the stable set polytope in one step has been open since 1990. We call these graphs $LS_+$-perfect. In the current contribution, we pursue a full combinatorial characterization of $LS_+$-perfect graphs and make progress towards such a characterization by establishing a new, close relationship among $LS_+$-perfect graphs, near-bipartite graphs and a newly introduced concept of full-support-perfect graphs.